Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 (For help, go the Skills Handbook, page 715.) Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . . Some are even and some are odd. 1. Make a list of the positive even numbers.  2. Make a list of the positive odd numbers.  3. Copy and extend this list to show the first 10 perfect squares. 12 = 1, 22 = 4, 32 = 9, 42 = 16, . . . 4. Which do you think describes the square of any odd number? It is odd. It is even. 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Solutions 1. Even numbers end in 0, 2, 4, 6, or 8: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, . . . 2. Odd numbers end in 1, 3, 5, 7, or 9: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, . . . 3. 12 = (1)(1) = 1; 22 = (2)(2) = 4; 32 = (3)(3) = 9; 42 = (4)(4) = 16; 52 = (5)(5) = 25; 62 = (6)(6) = 36; 72 = (7)(7) = 49; 82 = (8)(8) = 64; 92 = (9)(9) = 81; 102 = (10)(10) = 100 4. The odd squares in Exercise 3 are all odd, so the square of any odd number is odd. 1-1

Euclid Video

1-1 Patterns and Inductive Reasoning Objective: To use inductive reasoning to make conjectures. Inductive reasoning – reasoning that is based on patterns you observe

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Find a pattern for the sequence. Use the pattern to show the next two terms in the sequence. 384, 192, 96, 48, … Each term is half the preceding term. So the next two terms are 48 ÷ 2 = 24 and 24 ÷ 2 = 12. 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Make a conjecture about the sum of the cubes of the first 25 counting numbers. Find the first few sums. Notice that each sum is a perfect square and that the perfect squares form a pattern. 13 = 1 = 12 = 12 13 + 23 = 9 = 32 = (1 + 2)2 13 + 23 + 33 = 36 = 62 = (1 + 2 + 3)2 13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2 13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2 The sum of the first two cubes equals the square of the sum of the first two counting numbers. 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 The sum of the first three cubes equals the square of the sum of the first three counting numbers. (continued) This pattern continues for the fourth and fifth rows of the table. 13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2 13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2 So a conjecture might be that the sum of the cubes of the first 25 counting numbers equals the square of the sum of the first 25 counting numbers, or (1 + 2 + 3 + … + 25)2. 1-1

Counterexample – an example for which the conjecture is incorrect.

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 The first three odd prime numbers are 3, 5, and 7. Make and test a conjecture about the fourth odd prime number. One pattern of the sequence is that each term equals the preceding term plus 2. So a possible conjecture is that the fourth prime number is 7 + 2 = 9. However, because 3 X 3 = 9 and 9 is not a prime number, this conjecture is false. The fourth prime number is 11. 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 The price of overnight shipping was $8.00 in 2000, $9.50 in 2001, and $11.00 in 2002. Make a conjecture about the price in 2003. Write the data in a table. Find a pattern. 2000 $8.00 2001 2002 $9.50 $11.00 Each year the price increased by $1.50. A possible conjecture is that the price in 2003 will increase by $1.50. If so, the price in 2003 would be $11.00 + $1.50 = $12.50. 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Pages 6–9 Exercises 1. 80, 160 2. 33,333; 333,333 3. –3, 4 4. , 5. 3, 0 6. 1, 7. N, T 8. J, J 9. 720, 5040 10. 64, 128 11. , 1 16 32 36 49 12. , 13. James, John 14. Elizabeth, Louisa 15. Andrew, Ulysses 16. Gemini, Cancer 17. 18. 5 6 19. The sum of the first 6 pos. even numbers is 6 • 7, or 42. 20. The sum of the first 30 pos. 30 • 31, or 930. 21. The sum of the first 100 pos. even numbers is 100 • 101, or 10,100. 3 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 31. 31, 43 32. 10, 13 33. 0.0001, 0.00001 34. 201, 202 35. 63, 127 36. , 37. J, S 38. CA, CO 39. B, C 28. ÷ = and is improper. 29. 75°F 30. 40 push-ups; answers may vary. Sample: Not very confident, Dino may reach a limit to the number of push-ups he can do in his allotted time for exercises. 1 2 1 3 3 2 3 2 22. The sum of the first 100 odd numbers is 1002, or 10,000. 23. 555,555,555 24. 123,454,321 25–28. Answers may vary. Samples are given. 25. 8 + (–5 = 3) and 3 > 8 26. • > and • > 27. –6 – (–4) < –6 and –6 – (–4) < –4 31 32 63 64 / 1 3 1 2 1 3 1 3 1 2 1 2 / / 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 40. Answers may vary. Sample: In Exercise 31, each number increases by increasing multiples of 2. In Exercise 33, to get the next term, divide by 10. 41. You would get a third line between and parallel to the first two lines. 42. 43. 44. 45. 46. 102 cm 47. Answers may vary. Samples are given. a. Women may soon outrun men in running competitions. b. The conclusion was based on continuing the trend shown in past records. c. The conclusions are based on fairly recent records for women, and those rates of improvement may not continue. The conclusion about the marathon is most suspect because records date only from 1955. 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 48. a. b. about 12,000 radio stations in 2010 c. Answers may vary. Sample: Confident; the pattern has held for several decades. 49. Answers may vary. Sample: 1, 3, 9, 27, 81, . . . 1, 3, 5, 7, 9, . . . 52. 21, 34, 55 53. a. Leap years are years that are divisible by 4. b. 2020, 2100, and 2400 c. Leap years are years divisible by 4, except the final year of a century which must be divisible by 400. So, 2100 will not be a leap year, but 2400 will be. 50. His conjecture is probably false because most people’s growth slows by 18 until they stop growing somewhere between 18 and 22 years. 51. a. b. H and I c. a circle 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 54. Answers may vary. Sample: 100 + 99 + 98 + … + 3 + 2 + 1 1 + 2 + 3 + … + 98 + 99 + 100 101 + 101 + 101 + … + 101 + 101 + 101 The sum of the first 100 numbers is , or 5050. The sum of the first n numbers is . 55. a. 1, 3, 6, 10, 15, 21 b. They are the same. c. The diagram shows the product of n and n + 1 divided by 2 when n = 3. The result is 6. 100 • 101 2 n(n+1) 55. (continued) d. 56. B 57. I 58. [2] a. 25, 36, 49 b. n2 [1] one part correct 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 59. [4] a. The product of 11 and a three-digit number that begins and ends in 1 is a four-digit number that begins and ends in 1 and has middle digits that are each one greater than the middle digit of the three-digit number. (151)(11) = 1661 (161)(11) = 1771 b. 1991 c. No; (191)(11) = 2101 59. (continued) [3] minor error in explanation [2] incorrect description in part (a) [1] correct products for (151)(11), (161)(11), and (181)(11) 60-67. 68. B 69. N 70. G 1-1

Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Find a pattern for each sequence. Use the pattern to show the next two terms or figures. 1. 3, –6, 18, –72, 360 2. Use the table and inductive reasoning. 3. Find the sum of the first 10 counting numbers. 4. Find the sum of the first 1000 counting numbers. Show that the conjecture is false by finding one counterexample. 5. The sum of two prime numbers is an even number. –2160; 15,120 55 500,500 Sample: 2+3=5, and 5 is not even 1-1

Points, Lines, and Planes GEOMETRY LESSON 1-2 (For help, go to the Skills Handbook, page 722.) 1. y = x + 5 2. y = 2x – 4  3. y = 2x  y = –x + 7 y = 4x – 10 y = –x + 15 4. Copy the diagram of the four points A, B, C, and D. Draw as many different lines as you can to connect pairs of points. Solve each system of equations. 1-2

Points, Lines, and Planes GEOMETRY LESSON 1-2 1. By substitution, x + 5 = –x + 7; adding x – 5 to both sides results in 2x = 2; dividing both sides by 2 results in x = 1. Since x = 1, y = (1) + 5 = 6. (x, y) = (1, 6) 2. By substitution, 2x – 4 = 4x – 10; adding –4x + 4 to both sides results in –2x = –6; dividing both sides by –2 results in x = 3. Since x = 3, y = 2(3) – 4 = 6 – 4 = 2. (x, y) = (3, 2) 3. By substitution, 2x = –x + 15; adding x to both sides results in 3x = 15; dividing both sides by 3 results in x = 5. Since x = 5, y = 2(5) = 10. (x, y) = (5, 10) 4. The 6 different lines are AB, AC, AD, BC, BD, and CD. Solutions 1-2

. . . .

Which statement below is true? d. 0.235 > 2.354

1-2 Points, Lines, and Planes Objectives: To understand basic terms of geometry. To understand basic postulates of geometry.

1-2 Points, Lines, and Planes point – a location. A point has no size. It is represented by a small dot and is named by a capital letter. • C space – the set of all points line – a series of points that extends in two opposite directions ex AB “line AB” BA “line BA” or line t collinear points – points that lie on the same line

Points, Lines, and Planes GEOMETRY LESSON 1-2 In the figure below, name three points that are collinear and three points that are not collinear. Points Y, Z, and W lie on a line, so they are collinear. Any other set of three points do not lie on a line, so no other set of three points is collinear. For example, X, Y, and Z and X, W, and Z form triangles and are not collinear. 1-2

plane – a flat surface that has no thickness. A plane contains many lines and extends without end in all directions. coplanar – points and lines that are in the same plane.

EX Name four coplanar points. F, A, L, C G, E, O, B

Points, Lines, and Planes GEOMETRY LESSON 1-2 Name the plane shown in two different ways. You can name a plane using any three or more points on that plane that are not collinear. Some possible names for the plane shown are the following: plane RST plane RSU plane RTU plane STU plane RSTU 1-2

Plane EFG Plane FGH Plane HFE

postulate – an accepted statement of fact.

Points, Lines, and Planes GEOMETRY LESSON 1-2 Use the diagram below. What is the intersection of plane HGC and plane AED? As you look at the cube, the front face is on plane AEFB, the back face is on plane HGC, and the left face is on plane AED. The back and left faces of the cube intersect at HD. Planes HGC and AED intersect vertically at HD. 1-2

Plane EFB and Plane GFB

Points, Lines, and Planes GEOMETRY LESSON 1-2 Shade the plane that contains X, Y, and Z. Points X, Y, and Z are the vertices of one of the four triangular faces of the pyramid. To shade the plane, shade the interior of the triangle formed by X, Y, and Z. 1-2

D B

Points, Lines, and Planes GEOMETRY LESSON 1-2 9. Answers may vary. Sample: AE, EC, GA 10. Answers may vary. Sample: BF, CD, DF 11. ABCD 12. EFHG 13. ABHF 14. EDCG 15. EFAD Pages 13–16 Exercises 1. no 2. yes; line n 3. yes; line n 4. yes; line m 5. yes; line n 6. no 7. no 8. yes; line m 16. BCGH 17. RS 18. VW 19. UV 20. XT 21. planes QUX and QUV 22. planes XTS and QTS 23. planes UXT and WXT 24. UVW and RVW 1-2

Points, Lines, and Planes GEOMETRY LESSON 1-2 25. 26. 27. 28. 29. 30. S 31. X 32. R 33. Q 34. X 1-2

Points, Lines, and Planes GEOMETRY LESSON 1-2 46. Postulate 1-1: Through any two points there is exactly one line. 47. Answer may vary. Sample: 48. 49. not possible 35. no 36. yes 37. no 38. coplanar 39. coplanar 40. noncoplanar 41. coplanar 42. noncoplanar 43. noncoplanar 44. Answers may vary. Sample: The plane of the ceiling and the plane of a wall intersect in a line. 45. Through any three noncollinear points there is exactly one plane. The ends of the legs of the tripod represent three noncollinear points, so they rest in one plane. Therefore, the tripod won’t wobble. 1-2

Points, Lines, and Planes GEOMETRY LESSON 1-2 56. no 57. 58. yes 54. 55. 50. 51. not possible 52. 53. 1-2

Points, Lines, and Planes GEOMETRY LESSON 1-2 68. Answers may vary. Sample: Post. 1-3: If two planes intersect, then they intersect in exactly one line. 69. A, B, and D 70. Post. 1-1: Through any two points there is exactly one line. 59. yes 60. always 61. never 62. always 63. always 64. sometimes 65. never 66. a. 1 b. 1 c. 1 d. 1 e. A line and a point not on the line are always coplanar. 67. Post. 1-4: Through three noncollinear points there is exactly one plane. 1-2

Points, Lines, and Planes GEOMETRY LESSON 1-2 71. Post. 1-3: If two planes intersect, then they intersect in exactly one line. 72. The end of one leg might not be coplanar with the ends of the other three legs. (Post. 1-4) 73. yes 76. no 77. 74. 75. 1-2

Points, Lines, and Planes GEOMETRY LESSON 1-2 78. no 79. Infinitely many; explanations may vary. Sample: Infinitely many planes can intersect in one line. 80. By Post. 1-1, points D and B determine a line and points A and D determine a line. The distress signal is on both lines and, by Post. 1-2, there can be only one distress signal. 81. a. Since the plane is flat, the line would have to curve so as to contain the 2 points and not lie in the plane; but lines are straight. b. One plane; Points A, B, and C are noncollinear. By Post. 1-4, they are coplanar. Then, by part (a), AB and BC are coplanar. 82. 1 1-2

Points, Lines, and Planes GEOMETRY LESSON 1-2 91. I, K 92. 42, 56 93. 1024, 4096 94. 25, –5 95. 34 96. 44 83. 84. 1 85. A 86. I 87. B 88. H 89. [2] a. ABD, ABC, ACD, BCD b. AD, BD, CD [1] one part correct 90. The pattern 3, 9, 7, 1 repeats 11 times for n = 1 to 44. For n = 45, the last digit is 3. 1 4 1-2

Points, Lines, and Planes GEOMETRY LESSON 1-2 Use the diagram at right. 1. Name three collinear points. 2. Name two different planes that contain points C and G. 3. Name the intersection of plane AED and plane HEG. 4. How many planes contain the points A, F, and H? 5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line. D, J, and H planes BCGF and CGHD HE 1 Sample: Planes AEHD and BFGC never intersect. 1-2

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 (For help, go to Lesson 1-2.) Judging by appearances, will the lines intersect? 1. 2. 3. 4. the bottom 5. the top 6. the front 7. the back 8. the left side 9. the right side Name the plane represented by each surface   of the box. 1-3

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 Solutions 1. no 2. yes 3. yes 4-9. Answers may vary. Samples given: 4. NMR 5. PQL 6. NKL 7. PQR 8. PKN 9. LQR 1-3

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 Objectives: To identify segments and rays. To recognize parallel lines. 1-3

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 Segment – the part of a line consisting of two endpoints and all points between them Ray – the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint

opposite rays – two collinear rays with the same endpoint. Opposite rays always form a line.

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 Name the segments and rays in the figure. A ray is a part of a line consisting of one endpoint and all the points of the line on one side of that endpoint. A ray is named by its endpoint first, followed by any other point on the ray. So the rays are BA and BC. A segment is a part of a line consisting of two endpoints and all points between them. A segment is named by its two endpoints. So the segments are BA (or AB) and BC (or CB). The labeled points in the figure are A, B, and C. 1-3

• • • • A B C D a. Name the segments. b. Name the rays. • • • • A B C D a. Name the segments. b. Name the rays. c. Name the line in two different ways.

parallel lines – coplanar lines that do not intersect skew lines – noncoplanar lines that are not parallel and do not intersect.

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 Use the figure below. Name all segments that are parallel to AE. Name all segments that are skew to AE. Skew lines are lines that do not lie in the same plane. The four lines in the figure that do not lie in the same plane as AE are BC, CD, FG, and GH. Parallel segments lie in the same plane, and the lines that contain them do not intersect. The three segments in the figure above that are parallel to AE are BF, CG, and DH. 1-3

parallel planes – planes that do not intersect

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 Identify a pair of parallel planes in your classroom. Planes are parallel if they do not intersect. If the walls of your classroom are vertical, opposite walls are parts of parallel planes. If the ceiling and floor of the classroom are level, they are parts of parallel planes. 1-3

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 Pages 19-23 Exercises 1. 2. 3. 4. 5. RS, RT, RW, ST, SW, TW 6. RS, ST, TW, WT, TS, SR 7. a. TS or TR, TW b. SR, ST 8. 4; RY, SY, TY, WY 9. Answers may vary. Sample: 2; YS or YR, YT or YW 10. Answers may vary. Check students’ work. 11. DF 12. BC 13. BE, CF 14. DE, EF, BE 15. AD, AB, AC 16. BC, EF 17. ABC || DEF 1-3

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 18-20 Answers may vary. Samples are given 25. true 26. False; they are skew. 27. true 28. False; they intersect above CG. 29. true 30. False; they intersect above pt. A. 18. BE || AD 19. CF, DE 20. DEF, BC 21. FG 22. Answers may vary. Sample: CD, AB 23. BG, DH, CL 24. AF 31. False; they are ||. 32. False; they are ||. 33. Yes; both name the segment with endpoints X and Y. 34. No; the two rays have different endpoints. 35. Yes; both are the line through pts. X and Y. 1-3

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 36. 37. always 38. never 39. always 40. always 41. never 42. sometimes 43. always 44. sometimes 45. always 46. sometimes 47. sometimes 48. Answers may vary. Sample: (0, 0); check students’ graphs. 49. a. Answers may vary. Sample: northeast and southwest b. Answers may vary. Sample: northwest and southeast, east and west 50. Two lines can be parallel, skew, or intersecting in one point. Sample: train tracks–parallel; vapor trail of a northbound jet and an eastbound jet at different altitudes– skew; streets that cross–intersecting 1-3

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 55. a. The lines of intersection are parallel. b. Examples may vary. Sample: The floor and ceiling are parallel. A wall intersects both. The lines of intersection are parallel. 56. Answers may vary. Sample: The diamond structure makes it tough, strong, hard, and durable. The graphite structure makes it soft and slippery. 57. a. one segment; EF b. 3 segments; EF, EG, FG 51. Answers may vary. Sample: Skew lines cannot be contained in one plane. Therefore, they have “escaped” a plane. 52. ST || UV 53. Answers may vary. Sample: XY and ZW intersect at R. 54. Planes ABC and DCBF intersect in BC. 1-3

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 58. No; two different planes cannot intersect in more than one line. 59. yes; plane P, for example 60. Answers may vary. Sample: VR, QR, SR 61. QR 62. Yes; no; yes; explanations may vary. 63. D 64. H 65. B 66. F 67. B 68. C 69. D 57. c. Answers may vary. Sample: For each “new” point, the number of new segments equals the number of “old” points. d. 45 segments e. n(n – 1) 2 1-3

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 70. [2] a. Alike: They do not intersect. Different: Parallel lines are coplanar and skew lines lie in different planes. b. No; of the 8 other lines shown, 4 intersect JM and 4 are skew to JM. [1] one likeness, one difference 71–78. Answers may vary. Samples are given. 71. EF 72. A 73. C 74. AEF and HEF 75. ABH 76. EHG 77. FG 78. B 79. 80. 81. 82. 1.4, 1.48 83. –22, –29 84. FG, GH 85. P, S 86. No; whenever you subtract a negative number, the answer is greater than the given number. Also, if you subtract 0, the answer stays the same. 1-3

Points, Lines, and Planes GEOMETRY LESSON 1-2 Use the diagram at right. 1. Name three collinear points. 2. Name two different planes that contain points C and G. 3. Name the intersection of plane AED and plane HEG. 4. How many planes contain the points A, F, and H? D, J, and H planes BCGF and CGHD HE 1 1-2

Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 Use the figure below for Exercises 8 and 9. 8. Name a pair of parallel planes. 9. Name a line that is skew to XW. Use the figure below for Exercises 5-7. 5. Name the segments that form the triangle. 6. Name the rays that have point T as their endpoint. 7. Explain how you can tell that no lines in the figure are parallel or skew. RS, TR, ST plane BCD || plane XWQ TO, TP, TR, TS AC or BD The three pairs of lines intersect, so they cannot be parallel or skew. 1-3

Measuring Segments and Angles GEOMETRY LESSON 1-4 (For help, go to the Skills Handbook, pages 719 and 720.) Simplify each absolute value expression. 1. |–6| 2. |3.5| 3. |7 – 10| 4. |–4 – 2| 5. |–2 – (–4)| 6. |–3 + 12| 7. x + 2x – 6 = 6 8. 3x + 9 + 5x = 81 9. w – 2 = –4 + 7w Solve each equation. 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 Solutions 1. The number of units from 0 to –6 on the number line is 6. 2. The number of units from 0 to 3.5 on the number line is 3.5. 3. |7 – 10| = |–3|, and the number of units from 0 to –3 on the number line is 3. 4. |–4 – 2| = |–6|, and the number of units from 0 to –6 on the number line is 6. 5. |–2 – (–4)| = |–2 + 4| = |2|, and the number of units from 0 to 2 on the number line is 2. 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 Solutions (continued) 6. |–3 + 12| = |9|, and the number of units from 0 to 9 on the number line is 9. 7. Combine like terms: 3x – 6 = 6; add 6 to both sides: 3x = 12; divide both sides by 3: x = 4 8. Combine like terms: 8x + 9 = 81; subtract 9 from both sides: 8x = 72; divide both sides by 8: x = 9 9. Add –7w + 2 to both sides: –6w = –2; divide both sides by –6: w = 1 3 1-4

Solving Linear Equations EX 7 – 8x = 4x - 17 -4x -4x 7 – 12x = -17 -7 -7 -12x = -24 -12 -12 x = 2

Solving Linear Equations EX 4c + 3(c - 2) = -34 4c + 3(c) +3(-2) = -34 4c + 3c - 6 = -34 7c - 6 = -34 +6 +6 7c = -28 7 7 c = -4

1-4 Measuring Segments and Angles Objectives: To find the lengths of segments. To find the measures of angles.

1-4 Measuring Segments and Angles

congruent segments – two segments with the same length The symbol for congruence is

Measuring Segments and Angles GEOMETRY LESSON 1-4 Find which two of the segments XY, ZY, and ZW are congruent. Use the Ruler Postulate to find the length of each segment. XY = | –5 – (–1)| = | –4| = 4 ZY = | 2 – (–1)| = |3| = 3 ZW = | 2 – 6| = |–4| = 4 Because XY = ZW, XY ZW. 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 If AB = 25, find the value of x. Then find AN and NB. Use the Segment Addition Postulate to write an equation. AN + NB = AB Segment Addition Postulate (2x – 6) + (x + 7) = 25 Substitute. 3x + 1 = 25 Simplify the left side. 3x = 24 Subtract 1 from each side. x = 8 Divide each side by 3. AN = 2x – 6 = 2(8) – 6 = 10 NB = x + 7 = (8) + 7 = 15 Substitute 8 for x. AN = 10 and NB = 15, which checks because the sum of the segment lengths equals 25. 1-4

midpoint – a point that divides a segment into two congruent segments A midpoint, or any line, ray, or other segment through a midpoint is said to bisect the segment. Point B bisects line segment AC.

Measuring Segments and Angles GEOMETRY LESSON 1-4 M is the midpoint of RT. Find RM, MT, and RT. Use the definition of midpoint to write an equation. RM = MT Definition of midpoint 5x + 9 = 8x – 36 Substitute. 5x + 45 = 8x Add 36 to each side. 45 = 3x Subtract 5x from each side. 15 = x Divide each side by 3. RM = 5x + 9 = 5(15) + 9 = 84 MT = 8x – 36 = 8(15) – 36 = 84 Substitute 15 for x. RT = RM + MT = 168 RM and MT are each 84, which is half of 168, the length of RT. 1-4

angle – formed by two rays with the same endpoint

Measuring Segments and Angles GEOMETRY LESSON 1-4 Name the angle below in four ways. The name can be the vertex of the angle: G. Finally, the name can be a point on one side, the vertex, and a point on the other side of the angle: AGC, CGA. The name can be the number between the sides of the angle: 3. 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 EX Name all of the angles that have b as a vertex. C • A • D • B ABC CBD ABD 1-4

measuring angles

Measuring Segments and Angles GEOMETRY LESSON 1-4 Find the measure of each angle. Classify each as acute, right, obtuse, or straight. Use a protractor to measure each angle. m 1 = 110 Because 90 < 110 < 180, 1 is obtuse. m 2 = 80 Because 0 < 80 < 90, 2 is acute. 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 Suppose that m 1 = 42 and m ABC = 88. Find m 2. Use the Angle Addition Postulate to solve. m 1 + m 2 = m ABC Angle Addition Postulate. 42 + m 2 = 88 Substitute 42 for m 1 and 88 for m ABC. m 2 = 46 Subtract 42 from each side. 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 EX If the m RQT = 155, what are m RQS and m TQS. S • (3x + 14)° • T (4x – 20)° • • R Q m RQS + m SQT = m RQT Angle Addition Postulate. (4x – 20) + (3x + 14) = 155 7x – 6 = 155 7x = 161 x = 23 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 m RQS = 4x – 20 = 4(23) – 20 = 72 m SQT = 3x + 14 = 3(23) + 14 = 83 1-4

congruent angles – angles with the same measure

Measuring Segments and Angles GEOMETRY LESSON 1-4 9. 25 10. a. 13 b. RS = 40, ST = 24 11. a. 7 b. RS = 60, ST = 36, RT = 96 12. a. 9 b. 9; 18 13. 33 14. 34 1. 9; 9; yes 2. 9; 6; no 3. 11; 13; no 4. 7; 6; no 5. XY = ZW 6. ZX = WY 7. YZ < XW 8. 24 Pages 29–33  Exercises 15. 130 16. XYZ, ZYX, Y 17. MCP, PCM, C or 1 18. ABC, CBA 19. CBD, DBC 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 11. If RS = 8y + 4, ST = 4y + 8, and RT = 15y – 9, find the value of y. Then find RS, ST and RT. 8y + 4 4y + 8 • • • R S T RS + ST = RT (8y + 4) + (4y + 8) = 15y - 9 12y + 12 = 15y - 9 -3y = -21 y = 7 RS = 8y + 4 = 8(7) + 4 = 60 ST = 4y + 8 = 4(7) + 8 = 36 RT = 15y – 9 = 15(7) – 9 = 96 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 11. A is the midpoint of XY. 3x 5x - 6 • • • X A Y 3x = 5x - 6 -2x = - 6 x = 3 XA = 3x = 3(3) = 9 AY = 5x - 6 = 5(3) - 6 = 9 XY = 9 + 9 = 18 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 20-23. Drawings may vary. 20. 21. 22. 23. 33. –2.5, 2.5 34. –3.5, 3.5 35. –6, –1, 1, 6 36. a. 78 mi b. Answers may vary. Sample: measuring with a ruler 37–41. Check students’ work. 24. 60; acute 25. 90; right 26. 135; obtuse 27. 34 28. 70 29. Q 30. 6 31. –4 32. 1 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 60. 150 61. 30 62. 100 63. 40 64. 80 65. 125 66. 125 49. Answers may vary. Sample: (15, 0), (–9, 0), (3, 12), (3, –12) 50–54. Check students’ work. 55. about 42° 56–58. Answers may vary. Samples are given. 56. 3:00, 9:00 57. 5:00, 7:00 58. 6:00, 12:32 59. 180 42. true; AB = 2, CD = 2 43. false; BD = 9, CD = 2 44. false; AC = 9, BD = 9, AD = 11, and 9 + 9 11 45. true; AC = 9, CD = 2, AD = 11, and 9 + 2 = 11 46. 2, 12 47. 115 48. 65 = / 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 71. y = 15; AC = 24, DC = 12 72. ED = 10, DB = 10, EB = 20 73. a. Answers may vary. Sample: The two rays come together at a sharp point. b. Answers may vary. Sample: Molly had an acute pain in her knee. 74. 45, 75, and 165, or 135, 105, and 15 75. 12; m AOC = 82, m AOB = 32, m BOC = 50 76. 8; m AOB = 30, m BOC = 50, m COD = 30 77. 18; m AOB = 28, m BOC = 52, m AOD = 108 78. 7; m AOB = 28, m BOC = 49, m AOD = 111 79. 30 67–68. Answers may vary. Samples are given. 67. QVM and VPN 68. MNP and MVN 69. MQV and PNQ 70. a. 19.5 b. 43; 137 c. Answers may vary. Sample: The sum of the measures should be 180. 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 87. never 88. never 89. always 90. never 91. always 92. always 93. always 94. never 95. 25, 30 96. 3125; 15,625 97. 30, 34 80. a–c. Check students’ work. 81. Angle Add. Post. 82. C 83. F 84. D 85. H 86. [2] a. b. An obtuse measures between 90 and 180 degrees; the least and greatest whole number values are 91 and 179 degrees. Part of ABC is 12°. So the least and greatest measures for DBC are 79 and 167. [1] one part correct 1-4

Measuring Segments and Angles GEOMETRY LESSON 1-4 Use the figure below for Exercises 1-3. 1. If XT = 12 and XZ = 21, then TZ = 7. 2. If XZ = 3x, XT = x + 3, and TZ = 13, find XZ. 3. Suppose that T is the midpoint of XZ. If XT = 2x + 11 and XZ = 5x + 8, find the value of x. Use the figure below for Exercises 4–6. 4. Name 2 two different ways. 5. Measure and classify 1, 2, and BAC. 6. Which postulate relates the measures of 1, 2, and BAC? 9 DAB, BAD 24 90°, right; 30°, acute; 120°, obtuse Angle Addition Postulate 14 1-4

7. DE = 20. Point C is the midpoint of DE. Find CE. Basic Construction GEOMETRY LESSON 1-5 (For help, go to Lesson 1-3 and 1-4.) 1. CD 2. GH 3. AB 4. line m 5. acute ABC 6. XY || ST 7. DE = 20. Point C is the midpoint of DE. Find CE. 8. Use a protractor to draw a 60° angle. 9. Use a protractor to draw a 120° angle. In Exercises 1-6, sketch each figure. 1-5

1-6. Answers may vary. Samples given: Basic Construction GEOMETRY LESSON 1-5 Solutions 1-6. Answers may vary. Samples given: 1. The figure is a segment whose endpoints are C and D. 2. The figure is a ray whose endpoint is G. 3. The figure is a line passing through points A and B. 4. 5. The figure is an angle whose measure is between 0° and 90°. 6. The figure is two segments in a plane whose corresponding  lines are parallel. 1-5

Solutions (continued) Basic Construction GEOMETRY LESSON 1-5 7. Since C is a midpoint, CD = CE; also, CD + CE = 20; substituting results in CE + CE = 20, or 2CE = 20, so CE = 10. 8. 9. Solutions (continued) 1-5

Construct TW congruent to KM. Basic Construction GEOMETRY LESSON 1-5 Construct TW congruent to KM. Step 1: Draw a ray with endpoint T. Step 2: Open the compass to the length of KM. Step 3: With the same compass setting, put the compass point on point T. Draw an arc that intersects the ray. Label the point of intersection W. TW KM 1-5

Step 1: Draw a ray with endpoint Y. Basic Construction GEOMETRY LESSON 1-5 Construct Y so that Y G. Step 1: Draw a ray with endpoint Y. Step 2: With the compass point on point G, draw an arc that intersects both sides of G. Label the points of intersection E and F. 75° Step 3: With the same compass setting, put the compass point on point Y. Draw an arc that intersects the ray. Label the point of intersection Z. 1-5

Step 4: Open the compass to the length EF. Basic Construction GEOMETRY LESSON 1-5 (continued) Step 4: Open the compass to the length EF. Keeping the same compass setting, put the compass point on Z. Draw an arc that intersects the arc you drew in Step 3. Label the point of intersection X. Y G Step 5: Draw YX to complete Y. 1-5

perpendicular lines – two lines that intersect to form right angles. “perpendicular to”

Perpendicular bisector – a line, segment, or ray that is perpendicular to the segment at its midpoint. It bisects the segment into two congruent segments.

Step 1: Put the compass point on point A and draw a short arc. Make Basic Construction GEOMETRY LESSON 1-5 1 2 Use a compass opening less than AB. Explain why the construction of the perpendicular bisector of AB shown in the text is not possible. Start with AB. Step 1: Put the compass point on point A and draw a short arc. Make sure that the opening is less than AB. 1 2 Step 2: With the same compass setting, put the compass point on point B and draw a short arc. Without two points of intersection, no line can be drawn, so the perpendicular bisector cannot be drawn. 1-5

angle bisector – a ray that divides an angle into two congruent coplanar angles.

WR bisects AWB. m AWR = x and m BWR = 4x – 48. Find m AWB. Basic Construction GEOMETRY LESSON 1-5 WR bisects AWB. m AWR = x and m BWR = 4x – 48. Find m AWB. Draw and label a figure to illustrate the problem m AWR = m BWR Definition of angle bisector x = 4x – 48 Substitute x for m AWR and 4x – 48 for m BWR. –3x = –48 Subtract 4x from each side. x = 16 Divide each side by –3. m AWR = 16 m BWR = 4(16) – 48 = 16 Substitute 16 for x. m AWB = m AWR + m BWR Angle Addition Postulate m AWB = 16 + 16 = 32 Substitute 16 for m AWR and for m BWR. 1-5

Construct MX, the bisector of M. Basic Construction GEOMETRY LESSON 1-5 Construct MX, the bisector of M. Step 1: Put the compass point on vertex M. Draw an arc that intersects both sides of M. Label the points of intersection B and C. Step 2: Put the compass point on point B. Draw an arc in the interior of M. 1-5

Step 3: Put the compass point on point C. Basic Construction GEOMETRY LESSON 1-5 (continued) Step 3: Put the compass point on point C. Using the same compass setting, draw an arc in the interior of M. Make sure that the arcs intersect. Label the point where the two arcs intersect X. Step 4: Draw MX. MX is the angle bisector of M. 1-5

Basic Construction Pages 37-40 Exercises 9. a. 11; 30 1. 6. b. 30 GEOMETRY LESSON 1-5 Pages 37-40 Exercises 1. 2. 3. 4. 5. 9. a. 11; 30 b. 30 c. 60 10. 5; 50 11. 15; 48 12. 11; 56 13. 6. 7. 8. 1-5

16. Find a segment on XY so that you can construct YZ as its bisector. Basic Construction GEOMETRY LESSON 1-5 14. 15. 16. Find a segment on XY so that you can construct YZ as its bisector. 17. Find a segment on SQ so that you can construct SP as its bisector. Then bisect PSQ. 1-5

21. Explanations may vary. Samples are given. Basic Construction GEOMETRY LESSON 1-5 18. a. CBD; 41 b. 82 c. 49; 49 19. a-b. 20. Locate points A and B on a line. Then construct a at A and B as in Exercise 16. Construct AD and BC so that AB = AD = BC. 20. (continued) 21. Explanations may vary. Samples are given. a. One midpt.; a midpt. divides a segment into two segments. If there were more than one midpt. the segments wouldn’t be . 21. (continued) b. Infinitely many; there’s only 1 midpt. but there exist infinitely many lines through the midpt. A segment has exactly one bisecting line because there can be only one line to a segment at its midpt. c. There are an infinite number of lines in space that are to a segment at its midpt. The lines are coplanar. 1-5

They appear to meet at one pt. Basic Construction GEOMETRY LESSON 1-5 22. 23. 24. 25. They are both correct. If you mult. each side of Lani’s eq. by 2, the result is Denyse’s eq. 26. Open the compass to more than half the measure of the segment. Swing large arcs from the endpts. to intersect above and below the segment. Draw a line through the two pts. where the arcs intersect. The pt. where the line and segment intersect is the midpt. of the segment. 27. 28. a. They appear to meet at one pt. 1-5

c. The three bisectors of a intersect in one pt. Basic Construction GEOMETRY LESSON 1-5 33. a. b. They are all 60°. c. Answers may vary. Sample: Mark a pt., A. Swing a long arc from A. From a pt. P on the arc, swing another arc the same size that intersects the arc at a second pt., Q. Draw PAQ. To construct a 30° , bisect the 60° . 28. (continued) b. c. The three bisectors of a intersect in one pt. 29. 30. 31. impossible; the short segments are not long enough to form a . 32. impossible; the short segments are not long enough to form a . 1-5

c. Point O is the center of the circle. 36. ; the line intersects. Basic Construction GEOMETRY LESSON 1-5 34. a-c. 35. a-b. 35, (continued) c. Point O is the center of the circle. 36. ; the line intersects. 37. D 38. F 39. [2] a.Draw XY. With the compass pt. on B swing an arc that intersects BA and BC. Label the intersections P and Q, respectively. With the compass point on X, swing a arc intersecting XY. 39. [2] (continued) Label the intersection K. Open the compass to PQ. With compass pt. on K, swing an arc to intersect the first arc. Label the intersection R. Draw XR. 1-5

49. No; they do not have the same endpt. Basic Construction GEOMETRY LESSON 1-5 41. 6 42. 10 43. 4 44. 3 45. 46. 100 47. 20 and 180 48. 49. No; they do not have the same endpt. 50. Yes; they both represent a segment with endpts. R and S. 39. [2] b. With compass open to XK, put compass point on X and swing an arc intersecting XR. With compass on R and open to KR, swing an arc to intersect the first arc. Label intersection T. Draw XT. [1] one part correct 40. [4] a. Construct its bisector. b. Construct the bisector. Then construct the bisector of two new segments. 40. (continued) c. Draw AB. Do constructions as in parts a and b. Open the compass to the length of the shortest segment in part b. With the pt. of the compass on B, swing an arc in the opp. direction from A intersecting AB at C. AC = 1.25 (AB). [3] explanations are not thorough [2] two explanations correct [1] part (a) correct 1-5

1. Construct AC so that AC NB. Basic Construction GEOMETRY LESSON 1-5 NQ bisects DNB. 1. Construct AC so that AC NB. 2. Construct the perpendicular bisector of AC. 3. Construct RST so that RST QNB. 4. Construct the bisector of RST. 5. Find x. 6. Find m DNB. Use the figure at right. For problems 1-4, check students’ work. 17 88 1-5

Evaluate each expression for m = –3 and n = 7. The Coordinate Plane GEOMETRY LESSON 1-6 (For help, go to the Skills Handbook, pages 715 and 716.) Find the square root of each number. Round to the nearest tenth if necessary. 1. 25 2. 17 3. 123 4. (m – n)2 5. (n – m)2 6. m2 + n2 7. (a – b)2 8. 9. Evaluate each expression for m = –3 and n = 7. Evaluate each expression for a = 6 and b = –8. a + b 2 a2 + b2 1-6

The Coordinate Plane Solutions 1. 25 = 52 = 5 2. 17 4.1232 = 4.1 GEOMETRY LESSON 1-6 Solutions 1. 25 = 52 = 5 2. 17 4.1232 = 4.1 3. 123 11.0912 = 11.1 4. (m – n)2 = (–3 –7)2 = (–10)2 = 100 5. (n – m)2 = –7 – (–3))2 = (7 + 3)2 =102 = 100 6. m2 + n2 = (–3)2 + (7)2 = 9 + 49 = 58 7. (a – b)2 = (6 – (–8))2 = (6 + 8)2 =142 = 196 9. –2 2 = = –1 a + b 6 + (–8) 8. a2 + b2 = (6)2 + (–8)2 = 36 + 64 = 100 = 10 1-6

Find the distance between R(–2, –6) and S(6, –2) to the nearest tenth. The Coordinate Plane GEOMETRY LESSON 1-6 Find the distance between R(–2, –6) and S(6, –2) to the nearest tenth. Let (x1, y1) be the point R(–2, –6) and (x2, y2) be the point S(6, –2). d = (x2 – x1)2 + (y2 – y1)2 Use the Distance Formula. d = (6 – (–2))2 + (–2 – (–6))2 Substitute. d = 82 + (4)2 Simplify. d = 64 + 16 = 80 80 8.94427191 Use a calculator. To the nearest tenth, RS = 11.3. 1-6

How far is the subway ride from Oak to The Coordinate Plane GEOMETRY LESSON 1-6 How far is the subway ride from Oak to Symphony? Round to the nearest tenth. Oak has coordinates (–1, –2). Let (x1, y1) represent Oak. Symphony has coordinates (1, 2). Let (x2, y2) represent Symphony. d = (x2 – x1)2 + (y2 – y1)2 Use the Distance Formula. d = (1 – (–1))2 + (2 – (–2))2 Substitute. d = 22 + 42 Simplify. d = 4 + 16 = 20 20 4.472135955 Use a calculator. To the nearest tenth, the subway ride from Oak to Symphony is 4.5 miles. 1-6

( , ) The Coordinate Plane GEOMETRY LESSON 1-6 AB has endpoints (8, 9) and (–6, –3). Find the coordinates of its midpoint M. Use the Midpoint Formula. Let (x1, y1) be A(8, 9) and (x2, y2) be B(–6, –3). The midpoint has coordinates Midpoint Formula ( , ) x1 + x2 2 y1 + y2 Substitute 8 for x1 and (–6) for x2. Simplify. 8 + (–6) 2 The x–coordinate is = = 1 Substitute 9 for y1 and (–3) for y2. Simplify. 9 + (–3) 2 The y–coordinate is = = 3 6 The coordinates of midpoint M are (1, 3). 1-6

( , ) The Coordinate Plane GEOMETRY LESSON 1-6 The midpoint of DG is M(–1, 5). One endpoint is D(1, 4). Find the coordinates of the other endpoint G. Use the Midpoint Formula. Let (x1, y1) be D(1, 4) and the midpoint be (–1, 5). Solve for x2 and y2, the coordinates of G. ( , ) x1 + x2 2 y1 + y2 Find the x–coordinate of G. Find the y–coordinate of G. 4 + y2 2 5 = 1 + x2 –1 = Use the Midpoint Formula. –2 = 1 + x2 10 = 4 + y2 Multiply each side by 2. The coordinates of G are (–3, 6). 1-6

At the Bell A has coordinates (3, 8). B has coordinates (0, –4). C has coordinates (–5, –6). 1. Find the distance between A and B to the nearest tenth. 2. Find the midpoint M of AC to the nearest tenth. 12.4 (–1, 1) 1-6

The Coordinate Plane 11. about 4.5 mi 12. about 3.2 mi 13. 6.4 GEOMETRY LESSON 1-6 11. about 4.5 mi 12. about 3.2 mi 13. 6.4 14. 15.8 15. 15.8 16. 5 17. B, C, D, E, F 18. (4, 2) 19. (3, 1) 20. (3.5, 1) 21. (6, 1) 22. (–2.25, 2.1) 23. (3 , –3) 24. (10, –20) 25. (5, –1) 26. (0, –34) 27. (12, –24) 28. (9, –28) 29. (5.5, –13.5) 30. (8, 18) 1. 6 2. 18 3. 8 4. 9 5. 23.3 6. 10 7. 25 8. 12.2 9. 12.0 10. 9 mi Pages 46–49  Exercises 7 8 1-6

The midpts. Are the same, (5, 4). The diagonals bisect each other. 43. The Coordinate Plane GEOMETRY LESSON 1-6 31. (4, –11) 32. 5.0; (4.5, 4) 33. 5.8; (1.5, 0.5) 34. 7.1; (–1.5, 0.5) 35. 5.4; (–2.5, 3) 36. 10; (1, –4) 37. 2.8; (–4, –4) 38. 6.7; (–2.5, –2) 39. 5.4; (3, 0.5) 40. 2.2; (3.5, 1) 41. IV 42. The midpts. Are the same, (5, 4). The diagonals bisect each other. 43. ST = (5 – 2)2 + (–3 – (–6))2 = 9 + 9 = 3 2 4.2 TV = (6 – 5)2 + (–6 – (–3))2 = 1 + 9 = 10 3.2 VW = (5 – 6)2 + (–9 – (–6))2 = 9 + 9 = 3 2 3.2 SW = (5 – 2)2 + (–9 – (–6))2 = 9 + 9 = 3 2 4.2 No, but ST = SW and TV = VW. 1-6

53–56. Answers may vary. Samples are given. The Coordinate Plane GEOMETRY LESSON 1-6 50. 1073 mi 51. 2693 mi 52. 328 mi 53–56. Answers may vary. Samples are given. 53. (3, 6), (0, 4.5) 54. E (0, 0), (8, 4) 55. (1, 0), (–1, 4) 56. (0, 10), (5, 0) 44. 19.2 units; (–1.5, 0) 45. 10.8 units; (3, –4) 46. 5.4 units; (–1, 0.5) 47. Z; about 12 units 48. 165 units; The dist. TV is less than the dist. TU, so the airplane should fly from T to V to U for the shortest route. 49. 934 mi 57. exactly one pt., E (–5, 2) 58. exactly one pt., J (2, –2) 59. a–f. Answers may vary. Samples are given. a. BC = AD b. If two opp. sides of a quad. are both || and , then the other two opp. sides are . 1-6

c. The midpts. are the same. The Coordinate Plane GEOMETRY LESSON 1-6 59. (continued) c. The midpts. are the same. d. If one pair of opp. sides of a quad. are both || and , then its diagonals bisect each other. e. EF = AB 59. (continued) f. If a pair of opp. sides of a quad. are both || and , then the segment joining the midpts. of the other two sides has the same length as each of the first pair of sides. 60. A (0, 0, 0) B (6, 0, 0) C (6, –3.5, 0) D (0, –3.5, 0) E (0, 0, 9) F (6, 0, 9) G (0, –3.5, 9) 61. 62. 6.5 units 63. 11.7 units 64. B 65. I 1-6

b. Yes, R must be (–10, 8) so that RQ = 160. The Coordinate Plane GEOMETRY LESSON 1-6 66. A 67. C 68. A 69. [2] a. (–10, 8), (–1, 5), (8, 2) b. Yes, R must be (–10, 8) so that RQ = 160. [1] part (a) correct or plausible explanation for part (b) 70. 71. 72. 73. 74. 10 75. 10 76. 48 77. TAP, PAT 78. 150 1-6

1. Find the distance between A and B to the nearest tenth. The Coordinate Plane GEOMETRY LESSON 1-6 A has coordinates (3, 8). B has coordinates (0, –4). C has coordinates (–5, –6). 1. Find the distance between A and B to the nearest tenth. 2. Find BC to the nearest tenth. 3. Find the midpoint M of AC to the nearest tenth. 4. B is the midpoint of AD. Find the coordinates of endpoint D. 5. An airplane flies from Stanton to Mercury in a straight flight path. Mercury is 300 miles east and 400 miles south of Stanton. How many miles is the flight? 6. Toni rides 2 miles north, then 5 miles west, and then 14 miles south. At the end of her ride, how far is Toni from her starting point, measured in a straight line? 12.4 5.4 (–1, 1) (–3, –16) 500 mi 13 mi 1-6

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 (For help, go to the Skills Handbook page 719 and Lesson 1-6.) Simplify each absolute value. 1. |4 – 8| 2. |10 – (–5)| 3. |–2 – 6| 4. A(2, 3), B(5, 9) 5. K(–1, –3), L(0, 0) 6. W(4, –7), Z(10, –2) 7. C(–5, 2), D(–7, 6) 8. M(–1, –10), P(–12, –3) 9. Q(–8, –4), R(–3, –10) Find the distance between the points to the nearest tenth. 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 Solutions 1. | 4 – 8 | = | –4 | = 4 2. | 10 – (–5) | = | 10 + 5 | = | 15 | = 15 3. | –2 – 6 | = | –8 | = 8 4. d = (x2 – x1)2 + (y2 – y1)2 d = (5 – 2)2 + (9 – 3)2 d = 32 + 62 d = 9 + 36 = 45 To the nearest tenth, AB = 6.7. 5. d = (x2 – x1)2 + (y2 – y1)2 d = (0 – (–1))2 + (0 – (–3))2 d = 12 + 32 d = 1 + 9 = 10 To the nearest tenth, KL = 3.2. 6. d = (x2 – x1)2 + (y2 – y1)2 d = (10 – 4)2 + ( – 2 –(– 7))2 d = 62 + 52 d = 36 + 25 = 61 To the nearest tenth, WZ = 7.8. 7. d = (x2 – x1)2 + (y2 – y1)2 d = (– 7 – (– 5))2 + (6 – 2)2 d = (–2)2 + 52 d = 4 + 16 = 20 To the nearest tenth, CD = 4.5. 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 Solutions (continued) 8. 9. d = (x2 – x1)2 + (y2 – y1)2 d = (–12 – (–1))2 + (–3 – (–10))2 d = (–11)2 + 72 d = 121 + 49 = 170 To the nearest tenth, MP = 13.0. d = (–3 – (–8))2 + (–10 – (–4))2 d = 52 + (–6)2 d = 25 + 36 = 61 To the nearest tenth, QR = 7.8. 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 Margaret’s garden is a square 12 ft on each side. Margaret wants a path 1 ft wide around the entire garden. What will the outside perimeter of the path be? Because the path is 1 ft wide, increase each side of the garden by 1 ft. s = 1 + 12 + 1 = 14 P = 4s Formula for perimeter of a square P = 4(14) = 56 Substitute 14 for s. The perimeter is 56 ft. 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 G has a radius of 6.5 cm. Find the circumference of G in terms of . Then find the circumference to the nearest tenth. . C = 2 r Formula for circumference of a circle. C = 2 (6.5) Substitute 6.5 for r. C = 13 Exact answer. C = 13 40.840704 Use a calculator. The circumference of G is 13 , or about 40.8 cm. . 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 Quadrilateral ABCD has vertices A(0, 0), B(9, 12), C(11, 12), and D(2, 0). Find the perimeter. Draw and label ABCD on a coordinate plane. Find the length of each side. Add the lengths to find the perimeter. AB = (9 – 0)2 + (12 – 0)2 = 92 + 122 Use the Distance Formula. = 81 + 144 = 255 = 15 BC = |11 – 9| = |2| = 2 Ruler Postulate CD = (2 – 11)2 + (0 – 12)2 = (–9)2 + (–12)2 Use the Distance Formula. = 81 + 144 = 225 = 15 DA = |2 – 0| = |2| = 2 Ruler Postulate 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 (continued) Perimeter = AB + BC + CD + DA = 15 + 2 + 15 + 2 = 34 The perimeter of quadrilateral ABCD is 34 units. 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 To make a project, you need a rectangular piece of fabric 36 in. wide and 4 ft long. How many square feet of fabric do you need? Write both dimensions using the same unit of measurement. Find the area of the rectangle using the formula A = bh. 36 in. = 3 ft Change inches to feet using 12 in. = 1 ft. A = bh Formula for area of a rectangle. A = (4)(3) Substitute 4 for b and 3 for h. A = 12 You need 12 ft2 of fabric. 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 Find the area of B in terms of . . In B, r = 1.5 yd. . A = r2 Formula for area of a circle A = (1.5)2 Substitute 1.5 for r. A = 2.25 The area of B is 2.25 yd2. . 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 Find the area of the figure below. Draw a horizontal line to separate the figure into three nonoverlapping figures: a rectangle and two squares. 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 Find each area. Then add the areas. (continued) AR = bh Formula for area of a rectangle AR = (15)(5) Substitute 15 for b and 5 for h. AR = 75 AS = s2 Formula for area of a square AS = (5)2 Substitute 5 for s. AS = 25 A = 75 + 25 + 25 Add the areas. A = 125 The area of the figure is 125 ft2. 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 1. 22 in. 2. 36 cm 3. 56 in. 4. 78 cm 5. 120 m 6. 48 in. 7. 38 ft 8. 15 cm 9. 10 ft 10. 3.7 in. 11. m 12. 56.5 in. 13. 22.9 m 14. 1.6 yd 15. 351.9 cm 16. 14.6 units 17. 25.1 units Pages 55–58  Exercises 1 2 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 1 3 9 64 20. 1 ft2 or 192 in.2 21. 4320 in.2 or 3 yd2 22. 1 ft2 of 162 in.2 23. 8000 cm2 or 0.8 m2 24. 5.7 m2 or 57,000 cm2 25. 120,000 cm2 or 12 m2 26. 6000 ft2 or 666 yd2 27. 400 m2 28. 64 ft2 29. in.2 30. 0.25 m2 31. 9.9225 ft2 32. 0.01 m2 33. 153.9 ft2 34. 54.1 m2 35. 452.4 cm2 36. 452.4 in.2 37. 310 m2 38. 19 yd2 18. 16 units 19. 38 units 1 8 2 3 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 39. 24 cm2 40. 80 in.2 41. a. 144 in.2 b. 1 ft2 c. 144; a square whose sides are 12 in. long and a square whose sides are 1 ft long are the same size. 42. a. 30 squares b. 16; 9; 4; 1 c. They are =. Post 1-10 48. Answers may vary. Sample: For Exercise 46, you use feet because the bulletin board is too big for inches. You do not use yards because your estimated lengths in feet were not divisible by 3. 49. 16 cm 50. 96 cm2 51. 288 cm 43. 3289 m2 44–47. Answers may vary. Samples are given. 44. 38 in.; 90 in.2 45. 39 in.; 93.5 in.2 46. 12 ft; 8 ft2 47. 8 ft; 3.75 ft2 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 52. a. Yes; every square is a rectangle. b. Answers may vary. Sample: No, not all rectangles are squares. c. A = ( ) or A = 53. 512 tiles 56. 38 units 57. 54 units2 58. 1,620,000 m2 59. 30 m 60. (4x – 2) units 61. Area; the wall is a surface. 62. Perimeter; weather stripping must fit the edges of the door. 54. perimeter = 10 units area = 4 units2 55. perimeter = 16 units area = 15 units2 P 4 P2 16 2 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 b. c. 25 ft by 50 ft 63. Perimeter; the fence must fit the perimeter of the garden. 64. Area; the floor is a surface. 65. 6.25 units2 66. a. base height area 1 98   98 2 96  192 3 94  282 : 24 52 1248 25 50 1250 26 48 1248 47  6  282 48  4  192 49  2   98 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 67. a. 9 b. 9 c. 9 d. 9 68. units2 69. units2 70. (9m2 – 24mn + 16n2) units2 71. Answers may vary. Sample: one 8 in.-by-8 in. square + one 5 in.-by-5 in. square + two 4 in.-by-4 in. squares 72. 388.5 yd 73. 64 74. 2336 75. 540 76. 216 77. 810 78. (15, 13) 79. 8.5 units; (5.5, 5) 80. 5.8 units; (1.5, 5.5) 81. 13.9 units; (3, 5.5) 82. 6.4 units; (–2, 3.5) 83. 9.2 units; (1, 6.5) 84. 6.7 units; (–2.5, –2) 85. 90 86. WI RI 87. 62 units 88. 18 units 89. 6 units 90. 33 units 3a 20 25a2 4 1-7

Perimeter, Circumference, and Area GEOMETRY LESSON 1-7 1. Find the perimeter in inches. 2. Find the area in square feet. 3. The diameter of a circle is 18 cm. Find the area in terms of . 4. Find the perimeter of a triangle whose vertices are X(–6, 2), Y(8, 2), and Z(3, 14). 5. Find the area of the figure below. All angles are right angles. A rectangle is 9 ft long and 40 in. wide. 296 in. 30 ft2 81 cm2 42 units 256 in.2 1-7

1. Div. each preceding term by –2; , – Tools of Geometry GEOMETRY CHAPTER 1 Page 64 1. Div. each preceding term by –2; , – 2. Add 2 to the preceding term; 10, 12 3. Rotate the U clockwise one-quarter turn. Alphabet is backwards; 8. B 9. a. 1 b. infinitely many c. 1 d. 1 10. 29,054.0 ft2 11. never 12. sometimes 13. never 14. always 15. never 4. Answers may vary. Sample: 1, 2, 4, 8, 16, 32, . . . 1, 2, 4, 7, 11, 16, . . . In the first seq. double each term. In the second seq., add consecutive counting numbers. 5. A, B, C 6. Answers may vary. Sample: A, B, C, D 7. Answers may vary. Sample: A, B, D, E 1 2 4 1-A

Tools of Geometry 16. 10 17. a. (11, 19) b. MC = MD = 136 GEOMETRY CHAPTER 1 16. 10 17. a. (11, 19) b. MC = MD = 136 18. 19.1 units 19. 800 cm2 or 0.08 m2 20. 12.25 in.2 21. 63.62 cm2 22. 7 23. 9 24. Answers may vary. Sample: Some ways of naming an can help identify a side or vertex. 25. 26. bisector 27. VW 28. 7 units 1 3 29. AY 30. E, AY 31. 33 yd2 1-A

10. A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to a segment at its midpoint. 11. 12. 13. 14. 15. 16. 17.

18. 19. Name a pair of skew lines. AQ and BC 21. Name four noncoplanar points. A,Q, R, S 22. 23. 24. Always 25. Sometimes 26. Never 27. Never 28. Always 29. always

30. -7, 3 31. 0.5 32. 15 33. 31 34. 35.

38. 39. 40. 41. 42.

5. 6. 7. Name four noncoplanar points. A, B, D, E